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Rao-Blackwellised Particle Filter

Contents. Rao-Blackwell theoremMarginalizing the filter (Rao-Blackwellization)Generic Rao-Blackwellized Particle FilterPartially analytical state updatePractical RBPF AlgorithmMathematical Derivation. Rao-Blackwell theorem. Let d(X) be an estimator of an unobservable. X is the observable data.

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Rao-Blackwellised Particle Filter

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    1. Rao-Blackwellised Particle Filter

    2. Contents Rao-Blackwell theorem Marginalizing the filter (Rao-Blackwellization) Generic Rao-Blackwellized Particle Filter Partially analytical state update Practical RBPF Algorithm Mathematical Derivation

    3. Rao-Blackwell theorem Let d(X) be an estimator of an unobservable. X is the observable data. A sufficient statistic T(X) is an observable random variable such that the conditional probability distribution of all observable data X given T(X) does not depend on any of the unobservable quantities A Rao–Blackwell estimator d1(X) of an unobservable quantity ? is the conditional expected value E(d(X) | T(X)) of some estimator d(X) given a sufficient statistic T(X) The theorem states that the mean squared error of the Rao–Blackwell estimator does not exceed that of the original estimator

    4. Marginalizing the filter (Rao-Blackwellization) Particle filter target posterior: Suppose we divide state into two groups and such that and is analytically tractable Decompose posterior:

    5. Generic Rao-Blackwellized Particle Filter

    6. Partially analytical state update This step seems not contain the term .However this is an intractable theoretical formulation. If we try to decompose, we encounter expensive integrals over Mostly used method for this problem is approximating the conditional distribution over as a linear-Gaussian distribution.

    7. Practical RBPF Algorithm Assumptions: is independent of , conditioned on We can define are calculated from Particles are , is covariance matrix of

    8. Practical RBPF Algorithm

    9. Mathematical Derivation

    10. Mathematical Derivation

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